Stationary states with a certain energy. The special case when the Hamiltonian turns out to be independent of time is very important in practical terms. It corresponds to an action that does not explicitly depend on time (for example, when the potentials and do not contain time). In this case, the kernel does not depend on the time variable, but will be a function of only the interval. As a result of this fact, wave functions arise with periodic dependence from time to time.
The easiest way to understand how this happens is to look at a differential equation. Let's try to find a particular solution to the Schrödinger equation (4.14) in the form , that is, in the form of a product of a function that depends only on time and a function that depends only on coordinates. Substitution into equation (4.14) gives the relation
. (4.40)
The left side of this equation does not depend on , while the right side does not depend on . In order for this equation to be satisfied for any and , both of its parts must not depend on these variables, i.e., they must be constant. Let us denote such a constant by . Then
up to arbitrary constant factor. Thus, the required particular solution has the form
, (4.41)
where the function satisfies the equation
and this precisely means that the wave function corresponding to such a particular solution oscillates with a certain frequency. We have already seen that the frequency of oscillations of the wave function is related to classical energy. Therefore, when the wave function of a system has the form (4.41), then the system is said to have a certain energy. Each energy value corresponds to its own special function - a particular solution to equation (4.42).
The probability that the particle is at point is given by the square of the modulus of the wave function, i.e. By virtue of equality (4.41), this probability is equal and does not depend on time. In other words, the probability of detecting a particle at any point in space does not depend on time. In such cases, the system is said to be in a stationary state - stationary in the sense that the probabilities do not change in any way over time.
This stationarity is to some extent related to the uncertainty principle, since if we know that the energy is exactly equal to , time must be completely uncertain. This is consistent with our idea that the properties of an atom in a precisely defined state are completely independent of time, and if we measured we would get the same result at any moment.
Let be the energy value at which equation (4.42) has a solution , and let be another energy value corresponding to some other solution . Then we know two particular solutions to the Schrödinger equation, namely:
And 
; (4.43)
since the Schrödinger equation is linear, it is clear that along with its solution there will be and . In addition, if and are two solutions to an equation, then their sum is also a solution. Therefore it is clear that the function
will also be a solution to the Schrödinger equation.
In general, it can be shown that if all possible values energy and the corresponding functions are found, then any solution to equation (4.14) can be represented as a linear combination of all partial solutions of type (4.43) corresponding to certain energy values.
The total probability of finding a system at any point in space, as shown in the previous paragraph, is a constant. This should be true for any values of and . Therefore, using expression (4.44) for the function we obtain
(4.45)
Because right side must remain constant, then the time-dependent terms (i.e., terms containing exponents 
) must vanish regardless of the choice of coefficients and . This means that
. (4.46)
If two functions and satisfy the relation
then they are said to be orthogonal. Thus, from equality (4.46) it follows that two states with different energies are orthogonal.
Below we will give an interpretation of expressions like , and we will see that equality (4.46) reflects the fact that if a particle has energy [and, therefore, its wave function], then the probability of detecting a different energy value for it [i.e. e. wave function 
] must be equal to zero.
Problem 4.8. Show that when the operator is Hermitian, then the eigenvalue is real [for this we should put it in equality (4.30)].
Problem 4.9. Show the validity of equality (4.46) in the case when the operator is Hermitian [to do this, put , in equality (4.30).
Linear combinations of functions stationary states . Let us assume that the functions corresponding to the set of energy levels are not only orthogonal, but also normalized, i.e., the integral of the square of their modulus over all values equal to one:
, (4.47)
where is the Kronecker symbol defined by the equalities , if , and . Most functions known in physics can be represented as a linear combination orthogonal functions; in particular, any function that is a solution to the Schrödinger wave equation can be represented in this form:
. (4.48)
The odds are easy to find; multiplying expansion (4.48) by conjugate functions and integrating over , we obtain
(4.49)
and therefore
. (4.50)
Thus we got the identity
Another interesting way of getting the same result comes from the definition of the -function:
. (4.52)
The kernel can be expressed in terms of functions and energy values. We will do this using the following considerations. Let us be interested in what form the wave function has at the moment of time, if it is known to us at the moment of time. Since it is a solution to the Schrödinger equation, then for any it, like any solution of it, can be written in the form
. (4.53)
But at a moment in time. Previously, we expressed this as a relation that is actually equivalent to the integral over all values, i.e.
. (4.63)
Core for the case free particle will be written as
Problem 4.12. Calculate the integral (4.64) in quadratures. Show that the result is obtained in the form that the nucleus should actually have for a free particle [i.e. e. is a three-dimensional generalization of expression (3.3)].
TIME INDEPENDENT
TIME INDEPENDENT
(time-inconsistency) The peculiarity of a policy carried out over a certain period of time is that political choice depends on the commitments made in more early time. If political bodies have credibility, they can choose a policy that does not depend on the moment: for example, inflation in the current year can be reduced by making commitments to reduce government spending or about a decrease in the growth of the money supply next year. When it comes next year, the government may choose to fulfill its obligations rather than defer spending cuts to next year; the incentive to fulfill one's obligations is desirable to maintain the reputation that makes time-independent policies possible. Where political authorities do not have credibility, they have access only to policies that are appropriate at this moment time; There is no reason to take on obligations that no one trusts you to fulfill. See also: reputational policy.
Economy. Dictionary. - M.: "INFRA-M", Publishing House "Ves Mir". J. Black. General edition: Doctor of Economics Osadchaya I.M.. 2000 .
Economic dictionary. 2000 .
See what “TIME-INDEPENDENT” is in other dictionaries:
time dependent- - [V.A. Semenov. English-Russian dictionary on relay protection] Topics relay protection EN time dependent ...
time dependent parameter- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN time dependent parameter ... Technical Translator's Guide
time dependent availability factor- - Topics oil and gas industry EN time dependent availability… Technical Translator's Guide
time independent parameter- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy topics in general EN time independent parameter ... Technical Translator's Guide
Special, depending on general and individual mental and general personal properties this person a type of consciousness associated with the experience of time. Philosophical encyclopedic dictionary. 2010… Philosophical Encyclopedia
Time consciousness- a special type of consciousness, depending on many general and individual mental and personal properties of a given person, associated with the experience (perception) of time. The latter depends on the content of experiences and is mainly a possibility... The beginnings of modern natural science
time-based discipline- The order of servicing non-priority requests, depending on the time they remain in the system. If the service delay exceeds the set threshold, the request automatically becomes a priority. [L.M. Nevdyaev. Telecommunications... ... Technical Translator's Guide
- (from the Greek phasis appearance) period, stage in the development of a phenomenon, stage. Oscillation phase argument of the function describing the harmonic oscillatory process or an argument of a similar imaginary exponent. Sometimes it’s just an argument... ... Wikipedia
Or the beginning of Hamilton, in mechanics and mathematical physics serves to obtain differential equations movements. This principle applies to all material systems, whatever forces they may be subject to; First we will express it in that... Encyclopedic Dictionary F. Brockhaus and I.A. Ephron
Galileo's transformations into classical mechanics(Newtonian mechanics) transformation of coordinates and time when moving from one inertial system reference (ISO) to another. The term was coined by Philip Frank in 1909. Transformations... ...Wikipedia
The exact solution of the Schrödinger equation can be found only in a relatively small number of simple cases. Most problems in quantum mechanics lead to too complex equations, which cannot be solved exactly. Often, however, the conditions of the problem involve quantities of different orders; among them there may be small quantities, after neglecting which the problem is simplified so much that its exact solution becomes possible. In this case, the first step in solving the problem physical problem consists in the exact solution of the simplified problem, and the second is in the approximate calculation of the corrections due to small terms discarded in the simplified problem. General method to calculate these corrections is called perturbation theory.
Suppose that the Hamiltonian of a given physical system looks like
where V represents a small correction (perturbation) to the "unperturbed" operator. In §38, 39 we will consider perturbations V that do not explicitly depend on time (the same is assumed for ). The conditions necessary for the operator V to be considered “small” in comparison with the operator will be clarified below.
The perturbation theory problem for a discrete spectrum can be formulated as follows. It is assumed that the eigenfunctions and eigenvalues of the discrete spectrum of the unperturbed operator are known, that is, the exact solutions of the equation are known
It is required to find approximate solutions to the equation
i.e., approximate expressions for the eigenfunctions and values of the perturbed operator H.
In this section we will assume that all eigenvalues of the operator are non-degenerate. In addition, to simplify conclusions, we will first assume that there is only a discrete spectrum of energy levels.
It is convenient to carry out calculations from the very beginning in matrix form. To do this, let us decompose the required function into functions
Substituting this expansion into (38.2), we obtain
and multiplying this equality on both sides by and integrating, we find
Here we introduce the matrix of the perturbation operator V, defined using the unperturbed functions
We will look for the values of the coefficients and energy E in the form of series
where the quantities are of the same order of smallness as the disturbance V, the quantities are of the second order of smallness, etc.
Let us determine the corrections to the eigenvalue and own function, accordingly we assume: . To find the first approximation, we substitute into the equation, retaining only first-order terms. Equation c gives
Thus, the first approximation correction to the eigenvalue is equal to the average value of the disturbance in the state
Equation (38.4) with gives
a remains arbitrary and must be chosen so that the function is normalized up to first-order terms inclusive.
To do this, you need to put Indeed, the function
(the prime at the sum sign means that when summing over, the orthogonal term must be omitted and therefore the integral from differs from unity only by a value of the second order of smallness.
Formula (38.8) determines the first approximation correction to wave functions. From it, by the way, it is clear what the condition for the applicability of the method in question is. Namely, there must be inequality
that is, the matrix elements of the perturbation must be small compared to the corresponding differences in the unperturbed energy levels.
Let us also determine the correction of the second approximation to the eigenvalue. To do this, we substitute in (38.4) and consider terms of the second order of smallness. The equation gives
(we substituted ) from (38.7) and took advantage of the fact that, due to the Hermitian nature of the operator
Note that the second approximation correction to the energy normal condition always negative. Indeed, if it corresponds lowest value, then all terms in the sum (38.10) are negative.
Further approximations can be calculated in a similar way.
The results obtained are directly generalized to the case when the operator also has a continuous spectrum (and we're talking about still about the perturbed state of the discrete spectrum). To do this, you only need to add the corresponding integrals over the continuous spectrum to the sums over the discrete spectrum.
We will distinguish various states continuous spectrum with index v running through a continuous series of values; conventionally means a set of values of quantities sufficient for full definition states (if the states of the continuous spectrum are degenerate, which is almost always the case, then specifying the energy alone is not enough to determine the state). Then, for example, instead of (38,8) it will be necessary to write
and similarly for other formulas.
It is also useful to give a formula for the perturbed values of the matrix elements of any physical quantity calculated up to first order terms using functions from (38.8). It's easy to get the following expression)
In the first amount, and in the second.
Tasks
1. Determine the second approximation correction to the eigenfunctions.
Solution. We calculate the coefficients from equations (38.4) with , written up to second-order terms, and select the coefficient so that the function is normalized up to second-order terms. As a result we find
where we entered the frequencies
2. Determine the third approximation correction to eigenvalues energy.
Solution. Writing out terms of third order of smallness in equation (38.4), we obtain
3. Determine the energy levels of an anharmonic linear oscillator with a Hamiltonian
Solution. The matrix elements of x can be obtained directly according to the matrix multiplication rule, using expression (23.4) for the matrix elements of x. For non-zero matrix elements from we find
There are no diagonal elements in this matrix, so the first-order correction from the term in the Hamiltonian (considered as a perturbation to harmonic oscillator) absent. The correction of the second approximation from this term is of the same order as the correction of the first approximation from the term. The diagonal matrix elements from have the form
By using general formulas(38.6) and (38.10) we find as a result the following approximate expression for the energy levels of the anharmonic oscillator:
4. A spherical potential well with infinitely high walls undergoes a small deformation (without changing volume), taking the form of a weakly elongated or oblate ellipsoid of revolution with semi-axes and c. Find the splitting of the energy levels of a particle in a well under such deformation (A. B. Migdal, 1959).
Solution. Pit boundary equation
by replacing variables, it turns into the equation of a sphere of radius. By the same replacement, the Hamiltonian of the particle (M is the mass of the particle; the energy is measured from the bottom of the well) is transformed into , where
If you watch a movie from end to beginning, you'll probably get confused, but a quantum computer won't. This conclusion was reached by researcher Mile Gu from the center quantum technologies(Cqt) National University Singapore and Nanyang University of Technology, as well as other scientists.
In a study published in the journal Physical Review X, an international team of scientists shows that a quantum computer is less dependent on the “arrow of time” than a classical computer. In some cases, it seems that the quantum computer does not need to distinguish between cause and effect at all.
The new work is inspired by a discovery made almost 10 years ago by scientists James Crutchfield and John Mahoney at the University of California. They showed that many statistical data sequences will have a built-in arrow of time.
An observer who sees data played out from start to finish, like frames in a movie, can simulate what will happen next using only a modest amount of information about what happened before. An observer who tries to simulate a system in reverse gains much more difficult task– potentially needs to be tracked by an order of magnitude more information.
This discovery became known as causal asymmetry. It seems intuitive - after all, modeling a system when time goes by back seems like trying to deduce cause from effect. We used to find this more difficult than predicting the effect of a cause. IN everyday life Understanding what will happen next is easier if you know what just happened and what happened before.
However, researchers have always been intrigued to discover asymmetries associated with the ordering of time. This is because the fundamental laws of physics are ambiguous as to whether time moves forward or backward. “When physics does not impose any direction in time, where does causal asymmetry—the extra memory expenditure required to eliminate cause and effect—come from?” Gu asks.
The first studies of causal asymmetry used models with classical physics to generate predictions. Crutchfield and Mahoney teamed up with Gu and his colleagues to find out whether quantum mechanics situation.
And they found that it happened. Models using quantum physics, the team proves, can completely reduce the memory load. A quantum model forced to emulate a process in reverse time will always be superior to a classical model emulating a process in the future.
This work has a number of profound implications. “The most exciting thing for us is the possible connection with the arrow of time,” the scientists say. “If causal asymmetry occurs only in classical models, then this suggests that our perception of cause and effect, and hence time, may arise from the application of a classical explanation of events in a fundamentally quantum world.”
The most iconic is the thermodynamic arrow. This comes from the idea that disorder, or entropy, will always increase - a little here and there, in everything that happens, until the Universe ends up as one big, hot mess. Although causal asymmetry is not the same as the thermodynamic arrow, they may be related. Classic models that track more information also generate more clutter. Everything suggests that causal asymmetry may have entropic consequences.
